Department of Economics A Simple Model of Homophily in Social Networks Sergio Currarini, University of Leicester Jesse Matheson, University of Leicester Fernando Vega Redondo, Bocconi University Working Paper No. 16/05 A Simple Model of Homophily in Social Networks Sergio Currarini∗ Jesse Mathesony Fernando Vega Redondoz March 20, 2016 Abstract Biases in meeting opportunities have been recently shown to play a key role for the emergence of homophily in social networks (see Currarini, Jackson and Pin 2009). The aim of this paper is to provide a simple microfoundation of these biases in a model where the size and type- composition of the meeting pools are shaped by agents' socialization decisions. In particular, agents either inbreed (direct search only to similar types) or outbreed (direct search to population at large). When outbreeding is costly, this is shown to induce stark equilibrium behavior of a threshold type: agents \inbreed" (i.e. mostly meet their own type) if, and only if, their group is above certain size. We show that this threshold equilibrium generates patterns of in-group and cross-group ties that are consistent with empirical evidence of homophily in two paradigmatic instances: high school friendships and interethnic marriages. Keywords: Homophily, social networks, segregation. JEL Classification: D7, D71, D85, Z13. ∗Department of Economics, University of Leicester and Universita' Ca' Foscari di Venezia. Email: [email protected]. This author wishes to acknowledge the support of the Ministry of Education and Science of the Russian Federation, grant No. 14.U04.31.0002, administered through the NES CSDSI. yDepartment of Economics, University of Leicester. Email: [email protected]. zBocconi University and IGIER. Email: [email protected]. 1 1 Introduction A pervasive feature of social and economic networks is that contacts tend to be more frequent among similar agents than among dissimilar ones. This pattern, usually referred to as \homophily", applies to many types of social interaction, and along many dimensions of similarity.1 The presence of homophily has important implications on how information flows along the social network (see, for example, Golub and Jackson (2011)) and, more generally, on how agents' characteristics impinge on social behavior. It is therefore important to understand the generative process of homophilous social networks, and how agents' preferences and their meeting opportunities concur in determining the observed mix of social ties. The empirical evidence of many social networks shows that homophily is often in excess of the \baseline" level that would be expected under a uniform random assortment that reflected groups' population shares, and that inbreeding (within-group interaction) occurs both in small and large groups. In a seminal contribution, Currarini, Jackson and Pin (2009)|CJP henceforth| investigate the extent of such biases in the context of American high school friendships. They show that preferences which are biased in favor of same-type friendships help explain why member of large ethnic groups enjoy more popularity|a higher number of friends|than members of small ethnic groups. However, they find that the observed patterns of inbreeding homophily cannot be explained by a process in which agents meet purely at random. They conclude, therefore, that some kind of \meeting bias" must be at work.2 In this paper we study a micro-founded model of search which endogenously generates a meeting bias, and whose equilibrium predictions are consistent with the observed non-linear relationship between homophily and population shares common to both U.S. high school friendship nominations and U.S. marriages. A characterizing, and novel, feature of our model is the role of absolute group size in shaping agents incentives to either direct their search towards in-groups only, or to open up to interactions with out-groups as well. This marks a stark difference between our approach and the approach based on the role of population shares, central to all previous studies of homophily in economics and to Blau (1977)'s structural approach. We will discuss this difference in some detail in Sections 4 and 5. We test our model's predictions on the role of group size using micro data reflecting two different matching scenarios: friendship nominations and marriages. The empirical results support our model. We now present in some detail the model, and then discuss our contribution with respect to recent works on the subject. The essential features of our theoretical framework can be outlined as follows. Agents derive 1For an account of the pervasiveness of homophily, see the seminal work of Lazarsfeld and Merton (1954) and, more recently, Marsden (1987, 1988), Moody (2001), or the survey by McPearson, Smith-Lovin and Cook (2001). 2In particular, random meetings are shown to be inconsistent with the nonlinear relation between an index of homophily first proposed by Coleman (1958) and groups' population shares. 2 positive utility from the number of distinct ties they enjoy. Ties are formed from a fixed number of meeting draws obtained from an endogenously chosen meeting pool. Agents affect the composition of their meeting pool by choosing to either inbreed or outbreed. Inbreeding refers to the decision to restrict search to one's own group only; outbreeding refers to the decision to extend search to the whole population. The decision to inbreed or outbreed involves weighing conflicting incentives. Outbreeding is costly; we believe this reflects cultural, geographical, or linguistic barriers to access- ing other types. Inbreeding limits the size of the search pool and, therefore, the efficacy of search by affecting the probability of novel draws.3 An agent's breeding decision depends crucially on the size of her group, to the extent that this affects the probability of redundancies in search. Specifically, there exists a threshold group size above which the agent will inbreed and below which the agent will outbreed. We highlight two paradigmatic scenarios that embody polar assumptions on how agents connect. The first scenario involves a meeting mechanism where links and payoff flows are one-sided. This represents, for example, web-based social networks (such as Twitter) where links are directed and information flows in one direction. This scenario also captures, to some extent at least, the friendship nomination process on which the National Longitudinal Study of Adolescent Health is based. The second scenario involves a meeting mechanism where both connections and payoff flows are two-sided; links require some form of bilateral agreement or coordination. Marriages (mutual consent being required) are a natural example of this scenario. We show in Section 4 that the threshold equilibrium, together with some small random noise in meetings, predicts a qualitative pattern of the Coleman index which is consistent with the hump shaped pattern found in CJP for friendship nomination and also arising for U.S. marriages (see Figure 1 in the present paper). In addition, we show that focusing on the role of absolute group size helps explain differences in the aggregate homophily patterns of small vs. large schools that were identified in Currarini, Jackson and Pin (2010) but could not be explained in their framework (in large schools, the degree of homophily is uniformly higher). Using microlevel data on friendship nominations and marriages, in Section 5 we test other novel predictions of the model. One such prediction is that, conditional on relative population share and both in the one-sided and two-sided scenarios, inbreeding is more likely to occur in groups that are large in absolute size than in smaller groups. Another interesting theoretical prediction for which we find empirical support pertains, specifically, to the matching performance of small groups. It concerns the following contrast between one- and two-sided contexts. If matching is one-sided, the matches of any small group will have all other groups (large or small) represented according to their population shares. Instead, if matching is two-sided, the prediction is that outbreeders will meet each other with frequencies that reflect 3So, while in CJP the focus is on agents' decision of how intensively (i.e. for how long) to search for social ties, while the meeting probabilities are fixed exogenously (and hence outside of agents' influence), in our case all agents search with the same intensity, but are able to direct their search and thus affect their meeting probabilities. 3 their population shares within the pool of outbreeders. As a result, we have that outbreeding groups will be over-represented in the matches of other outbreeders relative to their population shares. We believe that all these findings provide strong empirical support to our model. The general idea that homophily patterns may stem from selection and assortative matching is present in many theoretical constructs and has been extensively tested empirically since Kandel's (1978) work on adolescent friendships. In Tiebout's \voting-by-feet" model, agents selectively structure their social interactions by forming homogeneous clubs along the preference dimension. The anticipation of future interaction is also at the heart of Baccara and Yariv (2013), where homophilous peer groups form in connected intervals along the preferences dimension. Selection may also result from information and opinion seeking, as in Suen (2010) mutual admiration clubs, where similar agents communicate in a sort of self confirming updating of information. Selection of agents with similar preferences may also stem from the desire to avoid strategic manipulation of information, as in Galeotti et al. (2013) model of cheap talk in networks. There is also a similarity between the main feature of our process (the difference in the cost of linking with in-group versus out-group agents) and the approach taken in Jackson and Roger (2005)'s islands model of network formation. However, while in that paper a key role is played by indirect benefits and the focus was on the emergence of small world architectures, here the focus is on the effect of different group sizes on homophily patterns in the absence of indirect benefit from connections.